1. Field of the Invention
The invention relates generally to the field of geophysical analysis of Earth formations and materials. More particularly, the invention relates to method for determining relative amounts of, and exponential decay parameters of Earth materials having one or more component, each of which has a unique exponential decay characteristic.
2. Background Art
Evaluation and analysis of Earth materials and Earth formations include various methods and apparatus for energizing the materials or formations with one or more types of energy, and measuring a response of the materials or formations to the energy. Some types of response are characterized by exponential decay, with respect to time, of a measured parameter related to the interaction of the Earth materials or formations with the particular form of energy. As two non-limiting examples of such measurements, nuclear magnetic resonance (NMR) spin echo amplitude is related with respect to time to the relative quantities of various components in the formations, each of which components can have a unique relationship of NMR spin-echo amplitude with respect to time. Such relationship is typically known as the transverse or longitudinal relaxation rate. The total NMR spin echo amplitude with respect to time measured in a particular material or formation is thus related to the quantity of each uniquely “decaying” component in the Earth material or formation being analyzed, and to the rate at which NMR spin echo amplitude decays with respect to time for each such component in the Earth material being analyzed. An apparatus for measuring NMR response in Earth formations is disclosed in U.S. Pat. No. 6,121,773 issued to Taicher et al., and incorporated herein by reference. See also U.S. Pat. No. 5,585,720 issued to Edwards.
Another example of exponentially decaying measurement parameter with respect to time, in response to energizing Earth materials, is known in the art as pulsed neutron decay measurement. In a system for measuring pulsed neutron response in an Earth material (or formation), controlled-duration impulses or “bursts” of high energy neutrons from a particular type of source are applied to the Earth materials being analyzed. Measurement are made with respect to time, at one or more locations spaced apart from the neutron source, of a response parameter related to the interaction of the high energy neutrons with the Earth materials. Examples of such measurements include capture gamma rays (gamma rays emitted when thermalized neutrons are captured by a susceptible nucleus in the Earth materials) or thermal neutrons. Various properties of the Earth materials can be inferred from the relative decay rates of the various components in the materials. See, for example, U.S. Pat. No. 5,789,752 issued to Mickael.
For the foregoing types of measurements, among others, determining the composition of the Earth materials requires determining the amplitude decay rate of each of the one or more components in the Earth materials being analyzed, as well as the fractional composition of each such component. Thus, the total signal amplitude (or parameter related to amplitude such as count rate in the case of neutron detection) can be represented as a sum of components, each having a unique decay rate and relative concentration (or fractional composition) in the Earth material being analyzed. Expressed mathematically:
      y    ⁡          (      t      )        =      b    +                  ∑                  i          =          1                N            ⁢                          ⁢                        A                      i            ⁢                                                                ⁢                  ⅇ                                    -                              k                i                                      ⁢            t                              
in which y(t) represents the signal amplitude at time t, Ai represents the relative concentration or quantity of the i-th component in the materials, and ki represents an exponential signal amplitude decay rate for the i-th component. b represents a background amplitude constant, which is typically the signal that would be measured after all the response effects of energizing the materials have decayed below an amplitude at which such effects are likely to be detected.
Methods are known in the art for determining the relative quantities of the various exponentially decaying components in a set of measurements relating to Earth material analysis. See the Edwards '720 patent mentioned above as one example. Methods known in the art typically require an explicit limit on the number of exponentially decaying components in a particular material being analyzed, require some initial estimate of the decay rate of the various components in the material, or both. What is needed is a generalized method for analyzing exponentially decaying signals from an Earth material that requires no initial estimates on the number of components or the decay rates of the various components.